Embodiments of the invention relate to signals, systems and methods such as, for example, modulation, navigation and positioning signals, systems methods and receivers adapted to receive and process the same.
Satellite Positioning Systems (SPS) rely on the passive measurement of ranging signals broadcast by a number of satellites, or ground-based or airborne equivalents, in a specific constellation or group of constellations. An on-board clock is used to generate a regular and usually continual series of events, often known as ‘epochs’, whose time of occurrence is coded into, or at least associated with, a random or pseudo-random code (known as a spreading code). As a consequence of the pseudo-random or random features of the time epoch encoding sequence, the spectrum of the output signal is spread over a frequency range determined by a number of factors including the rate of change of the spreading code elements and the waveform used for the spreading signal. In the prior art, the spreading waveform is rectangular, of constant chipping rate, and has a (sinc)2 function power spectrum, prior to filtering by transmission circuitry.
The ranging signals are modulated onto a carrier signal for transmission to passive receivers. Applications are known that cover land, airborne, marine and space use. Typically, binary phase shift keying is employed to modulate the carrier signal, which, itself, has a constant magnitude. Usually, at least two such signals are modulated onto the same carrier in phase quadrature. The resulting carrier signal retains its constant envelope but has four phase states depending upon the two independent input signals. However, it will be appreciated that the two modulating signals do not need to have the same carrier magnitude. It is possible for a constant carrier magnitude of the combined signal to be maintained by appropriate selection of corresponding phases other than π/2 radians.
Techniques are known by which more than two signals are modulated onto the same carrier using either additive methods (known as ‘Interplex’ modulation) or a combination of angle modulation and additive methods, known as ‘Coherent Adaptive Sub-carrier Modulation’ (CASM). Both of these techniques require the addition of a further inter-modulation component that is derived to maintain constant carrier magnitude. For example, in ‘Interplex’ modulation, techniques are known with three transmitted components, 2 on one carrier phase with a third on the quadrature phase. These have at least six phase states.
An example of such a satellite positioning system is the Global Positioning System (GPS). Generally, the GPS operates using a number of frequencies such as, for example, L1, L2 and L5, which are centred at 1575.42 MHz, 1227.6 MHz and 1176.45 MHz respectively. Each of these signals is modulated by respective spreading signals. As will be appreciated by those skilled in the art, a Coarse Acquisition (CA) code signal emitted by the GPS Satellite Navigation System is broadcast on the L1 frequency of 1575.42 MHz with a spreading code rate (chip rate) of 1.023 MHz. The CA code signal has a rectangular spreading waveform, is binary phase shift keyed on to the carrier, and is categorised as BPSK-R1. The GPS signal structure is such that the signal broadcast by the satellites on the L1 frequency has a second component in phase quadrature, which is known as the precision code (P(Y) code) and made available to authorised users only. The P(Y) signal is BPSK modulated with a spreading code at 10.23 MHz with a magnitude that is 3 dB lower in signal power than the CA code transmission. Consequently, the Q component has a magnitude that is 0.7071 (−3 dB) of the magnitude of the I component. It will be appreciated by those skilled in the art that the phase angles of these states of these signals are ±35.265° in relation to the ±1 axis (phase of the CA code signal as specified in ICD GPS 200C). One skilled in the art also appreciates that the P code is a function of or is encrypted by the Y code. The Y code is used to encrypt the P code. One skilled in the art appreciates that the L1 signal, containing both I & Q components, and the L2 signal can be represented, for a given satellite, i, asSL1i(t)=APpi(t)di(t)cos(ω1t)+ACci(t)di(t)sin(ω1t), andSL2i(t)=BPpi(t)di(t)cos(ω2t)
where
AP and AC are the amplitudes of the P and CA codes, typically AP=2AC;
Bp is the amplitude of the L2 signal;
ω1 and ω2 are the L1 and L2 carrier frequencies;
pi(t) represents the P(Y) ranging code and is a pseudo-random sequence with a chip rate of 10.23 Mcbps. The P code has a period of exactly 1 week, taking values of +1 and −1;
ci(t) represents the CA ranging code and is a 1023 chip Gold code, taking values of +1 and −1; and
di(t) represents the data message, taking values of +1 and −1.
In the near future, it is expected that a third military signal, designated M-code, will be transmitted in the L1 band by GPS satellites.
A satellite constellation typically comprises 24 or more satellites often in similar or similarly shaped orbits but in a number of orbital planes. The transmissions from each satellite are on the same nominal carrier frequency in the case of code division access satellites (such as GPS) or on nearby related frequencies such as GLONASS. The satellites transmit different signals to enable each one to be separately selected even though several satellites are simultaneously visible.
The signals from each satellite, in a CDMA system like GPS, are distinguished from one another by means of the different spreading codes and/or differences in the spreading code rates, that is, the pi(t) and ci(t) sequences. Nevertheless, there remains significant scope for interference between the signals transmitted by the satellites. Typically, the power spectrum for the CA code has maximum power at the carrier frequency L1 and zeros at multiples of the fundamental frequency, 1.023 MHz, of the CA code. Therefore, it will be appreciated that zeros occur either side of the carrier frequency at ±1.023 MHz, ±2.046 MHz etc. Similarly, the power spectrum for a the P(Y) code has a maximum amplitude centred on the L1 and L2 frequencies, with zeros occurring at multiples of ±10.23 MHz as is expected with a sinc function waveform.
It is known to further modulate the ranging codes using a sub-carrier, that is, a further signal is convolved with signals similar to the P codes and/or CA codes, to create Binary Offset Carrier (BOC) modulation as can be appreciated from, for example, J. W. Betz, “Binary Offset Carrier Modulation for Radionavigation”, Navigation, Vol. 48, pp 227-246, Winter 2001-2002, International patent application PCT/GB2004/003745 and “Performance of GPS Galileo Receivers Using m-PSK BOC Signals”, Proceedings of Institute of Navigation Conference, 2003. 9-12 Sep. 2003, Portland, Oreg., USA, Pratt, A. R., Owen J. I. R. all of which are incorporated herein by reference. Standard BOC modulation is well-known. The combination of a portion of a binary spreading code with a binary subcarrier signal produces the BOC signal used to modulate a carrier such as, for example, L1. The BOC signal is formed by the product of a binary sub-carrier (known as the spreading symbol modulation), which is rectangular square wave, and the spreading symbols (the sequence of spreading code elements). The BOC spreading symbol modulation can be represented as, for example, ci(t)*sign(sin(2πfst)), where fs is the frequency of the subcarrier. One skilled in the art understands that BOC(fs,fc) denotes Binary Offset Carrier modulation with a subcarrier frequency of fs and a code rate (or chipping rate) of fc. Using binary offset carriers results in the following exemplary signal descriptions of the signals emitted from the satellite:SL1i(t)=Amscim(t)mi(t)di(t)cos(ω1t)+Agscig(t)gi(t)di(t)sin(ω1t)=ISL1i(t)+QSL1i(t), andSL2i(t)=Bmscim(t)mi(t)di(t)cos(ω2t)
where
Am, Ag and Bm are amplitudes;
mi(t) is the spreading code for the in-phase (cosine) component of the signal;
gi(t) is the spreading code for the quadrature (sine) component of the signal;
scim(t) represents the sub-carrier signal for mi(t);
scig(t) represents a subcarrier signal for gi(t);
ω1 and ωt are designated as L1 and L2 carrier frequencies.
It will be appreciated that the embodiment expressed above uses a single component on the in-phase and a single component on the quadrature phase for the L1 signal. Similarly, the L2 signal comprises a single component. However, one skilled in the art appreciates that the L1 and/or L2 signals may use one or more components.
BOC signals are typically rectangular or square waves. However, alternatives have been proposed that involve more complex spreading symbol modulation utilising multiple signal levels as can be appreciated from, for example, International patent application PCT/GB2004/003745, and “Performance of GPS Galileo Receivers Using m-PSK BOC Signals”, Proceedings of Institute of Navigation Conference, 2003. 9-12 Sep. 2003, Portland, Oreg., USA, Pratt, A. R., Owen J. I. R cited above. These provide a means for better control of the resulting signal spectrum as the power spectral density Φn,m(x), where x is a generalised frequency variable, of a BOC spreading symbol modulation is fully defined by the equation:
            Φ              n        ,        m              ⁡          (      x      )        =                    2        ⁢                                  ⁢        π                    m        ⁢                                  ⁢                  ω          0                      ·                  {                                            sin              ⁡                              (                                                      π                    ⁢                                                                                  ⁢                    x                                                        2                    ⁢                                                                                  ⁢                    n                                                  )                                      ·                          sin              ⁡                              (                                                      π                    ⁢                                                                                  ⁢                    x                                    m                                )                                                                        {                                                π                  ⁢                                                                          ⁢                  x                                m                            }                        ·                          cos              ⁡                              (                                                      π                    ⁢                                                                                  ⁢                    x                                                        2                    ⁢                                                                                  ⁢                    n                                                  )                                                    }            2                      where x=ω/ω0         
In a subset of the multi-level digital waveforms used as spreading symbol modulation waveforms, a specific category has been recognised that has attracted the name Composite BOC (CBOC) as can be appreciated from, for example, Avila-Rodriguez, J. A. et al, “Revised Combined Galileo/GPS Frequency and Signal Performance Analysis”, Proceedings of Institute of Navigation Conference, 2005, 13-16 Sep. 2005, Long Beach, Calif., USA, which is incorporated herein by reference for all purposes, in which several Binary Offset Carrier signals are additively combined to form the spreading symbol modulation waveform.
A further option for spectrum control has also arisen that uses time-multiplexed techniques in which several BOC spreading symbol modulation waveforms are combined in a defined time sequence as can be appreciated from the above PCT application and Pratt, A. R., Owen, J. I. R., “Signal Multiplex Techniques in Satellite Channel Availability, Possible Applications to Galileo”, GNSS 2005, Institute of Navigation Conference Record, pp 2448-2460, Sep. 13-16, 2005, Long Beach and Pratt, A. R. Owen, J. I. R., “Galileo Signal Optimisation in L1”, Conference Record, National Technical Meeting, Institute of Navigation, pp 332-345, Jan. 24-26, 2005, San Diego. This technique assigns a specific spreading symbol modulation, drawn from a defined alphabet of such modulation waveforms, one to every spreading code element (or time slot—quantised by code element). Through the process of selecting which BOC modulation is used in which time slot, the relative proportions of each spreading symbol modulation component can be controlled. Only binary versions of this arrangement are known although it will be clear to those skilled in the art that multi-level equivalent arrangements are also possible that involve both time multiplexed techniques to determine which spreading symbol modulation is used in each time slot and the use of an alphabet of spreading symbol modulations that are multi-level and may be a combination of basic BOC spreading symbol waveforms. Such combinations may be in exemplary realisations either additive or multiplicative or some other means for combining the base modulation waveforms.
Multiplexed BOC
A proposal has been made for several satellite navigation systems to use a common modulation spectrum so that the signals/services maintain a degree of interoperability as can be appreciated from, for example, Hein, G. W. et al, MBOC: The New Optimized Spreading Modulation for GALILEO L1 OS and GPS L1C, Conference Record, IEEE PLANS/IoN National Technical Meeting, San Diego, April 2006, Session C5 Paper 7. The common spectrum does not require different satellite navigation systems to emit waveforms that are identical. The disclosed common spectrum, known as multiplex BOC or MBOC, may be attained by either a time multiplex technique or by the superposition (addition) of the required BOC components. The time multiplex technique, using binary offset carriers, has become known as TMBOC, whilst the superposition technique has become known as composite BOC, or by its nitial letters, CBOC.
The time multiplex method of constructing a spreading symbol modulation waveform using BOC modulation components is illustrated in FIG. 2, which shows a pair of signals 200. An overall BOC signal or subcarrier 202 comprises a number 204 to 208 of bursts of a first spreading symbol modulation A, each burst of which has the duration of one chip of the spreading code. There may be several successive chips with this modulation. The overall MBOC 202 also comprises at least one burst 210 of a second, distinct, spreading symbol modulation B with similar characteristics but having a different carrier offset frequency. The depicted MBOC 202 also comprises a third spreading symbol modulation burst 212, which is identified as modulation type C with yet a further carrier offset frequency. In the known art, each of these modulation bursts has a BOC characteristic but with a common chip rate. Prior to transmission from a navigation satellite, the carrier signal and spreading symbol modulation are further modulated by a spreading code 214. It can be appreciated that only an exemplary number of chips, chip n to chip n+4, of the complete spreading code are illustrated. For the time multiplex technique with binary offset carrier spreading symbol modulation components, the relative magnitude of the components is determined by the proportion of time (in units of code sequence elements or chips) devoted to each. In the example of FIG. 2, the proportion allotted to the first spreading symbol modulation A is ⅗, to B is ⅕ and to C is ⅕, provided that this pattern were to continue ad infinitum. It will be clear to those skilled in the art that other proportions are possible within the restriction that the relative power of each component is set in multiples of 1/N, where N is the length of the repetitive spreading sequence. This restriction can be overcome also by having different time multiplex assignments for each repetition of the spreading sequence.
CBOC
The alternative formulation of the MBOC spectrum is by means of an additive method, whereby two time-continuous binary offset carrier spreading symbol modulation waveforms are additively combined. FIG. 3 provides an illustration 300 of the waveform produced using this method. First 302 and second 304 BOC components or waveforms are illustrated. The relative magnitudes of the two components 302 and 304 are controlled through the amplitudes of each of the BOC components. The first 302 BOC is the base-line BOC waveform, which is a BOC(1,1) waveform. The second waveform 304 illustrated a BOC(5,1) waveform. A number of chips, chip n to chip n+4, of a spreading code 306 is illustrated. The CBOC waveform 308 resulting from the additive combination of the first and second waveforms 302 and 304 is shown. It can be appreciated that CBOC waveform 308 comprises first and second components reflecting, respectively, their relationship to the first 302 and second 304 BOCs. The second component 310 is reduced in magnitude compared with the first component. For the 2 component CBOC waveform 308 shown, the resulting signal waveform has 4 levels. In general, a CBOC waveform has 2n levels when derived from n BOC waveforms. Depending upon the relative amplitudes, it is possible that some of these levels may coincide.
Binary Offset Carrier Spreading Symbol Modulation
The conventional means of identifying the characteristics of binary offset carrier modulation is through 2 parameters n and m. The modulation is denoted BOC(n,m), in which n applies to the frequency of the offset carrier and m refers to the chipping rate. The parameters m and n are usually associated with a GPS-like signal in which the master satellite clock oscillates at 10.23 MHz or some multiple or fraction thereof. The parameters may then take on the meanings expressed by:Offset carrier frequency=n×1.023 MHzChipping rate=m×1.023 M chips per second.
In the known implementation of a time multiplexed spectrum containing the two BOC modulation components, it is known that the phase of the spreading symbol modulation is identical at the transition to each code element (chip). For example, if the BOC spreading symbol modulation has a positive transition at the beginning of a specific code element, having the value +1, and a negative transition at the beginning of a specific code element, having the value −1, then these phase assignments may be applied to each spreading symbol in the complete sequence.
MBOC
One common power spectral density (PSD) that might be used by both Galileo and GPS navigation constellations is:
                                          Φ            MBOC                    ⁡                      (            ω            )                          =                                            10              11                        ·                                          Φ                                  (                                      1                    ,                    1                                    )                                            ⁡                              (                ω                )                                              +                                    1              11                        ·                                          Φ                                  (                                      6                    ,                    1                                    )                                            ⁡                              (                ω                )                                                                        (        1        )            
In many satellite navigation systems, it is normal to transmit both a data-bearing signal and a ‘so-called’ pilot signal, which does not carry a data message. The data message is transmitted at a lower rate than the spreading code. For GPS CA code, the spreading code rate is 1.023 MHz whilst the data message is transmitted at 50 bits per second. In modernised GPS, both the pilot and data signals are transmitted although not necessarily at the same power levels. In the time multiplexed method of generating the MBOC spectrum, there are a wide range of options for choosing assignments for the division of power between the pilot and data channels. This permits the option of transmitting different relative proportions of power for each of the BOC spreading symbol components on the pilot and data-bearing signals. For example, if the two spreading symbol modulation components are BOC(1,1) and BOC(6,1), then the data-bearing signal, carrying a proportion γ of the total power, uses the BOC(1,1) spreading symbol modulation only whilst the pilot signal, carrying a proportion (1−γ) of the total power, would use a time multiplexed version having the power spectral density:
                                                        Φ              Pilot                        ⁡                          (              ω              )                                =                                                    (                                                      10                    11                                    -                  γ                                )                            ·                              (                                  1                                      1                    -                    γ                                                  )                            ·                                                Φ                                      (                                          1                      ,                      1                                        )                                                  ⁡                                  (                  ω                  )                                                      +                                          1                11                            ⁢                                                (                                      1                                          1                      -                      γ                                                        )                                ·                                                      Φ                                          (                                              6                        ,                        1                                            )                                                        ⁡                                      (                    ω                    )                                                                                      ⁢                                  ⁢                                  ⁢                                            Φ              Data                        ⁡                          (              ω              )                                =                                    Φ                              (                                  1                  ,                  1                                )                                      ⁡                          (              ω              )                                                          (        2        )            
This arrangement allows considerable freedom in selecting the proportions of power allocated to the data-bearing and pilot signals and in determining how the two BOC(n,m) components are distributed between these two signals. Equation (2) maintains the combined PSD for both pilot and data-bearing signals in accordance with the required MBOC PSD.
CBOC
For the Composite BOC method, the selection of parameters to provide for power division is more complex.
The equations that follow show the complexity associated with the control of the CBOC power spectral density. It is assumed that there are at least two components in the composite BOC spectrum. For illustrative purposes, the equations below are constructed for 2 components. However, those skilled in the art will recognise that more than 2 components may be used.
The spectrum of a binary offset carrier, BOC(n,m), with a sine phased spreading symbol modulation, is given in equation (3). Equation (3) shows the complex spectrum, Hn,m(ω), for values of (2n/m) that are even. This corresponds to (n/m) complete cycles of the binary offset carrier in each spreading code symbol. The complex spectrum is based on a calculation over the duration of a single code element, ΔT=2π/(mω0). The waveform used for the spectrum computations extends over the interval t∈(−ΔT/2, ΔT/2) and, for definition, has a positive transition at t=0.
                                          H                          n              ,              m                        sin                    ⁡                      (            ω            )                          =                                            2              ⁢                                                          ⁢                              π                ·                                                      (                                          -                      1                                        )                                                        (                                                                  n                        m                                            +                      1                                        )                                                                                      j              ⁢                                                          ⁢              m              ⁢                                                          ⁢                              ω                0                                              ·                                                    sin                ⁡                                  (                                                            π                      ⁢                                                                                          ⁢                      x                                                              2                      ⁢                                                                                          ⁢                      n                                                        )                                            ·                              sin                ⁡                                  (                                                            π                      ⁢                                                                                          ⁢                      x                                        m                                    )                                                                                    (                                                      π                    ⁢                                                                                  ⁢                    x                                    m                                )                            ·                              cos                ⁡                                  (                                                            π                      ⁢                                                                                          ⁢                      x                                                              2                      ⁢                                                                                          ⁢                      n                                                        )                                                                                        (        3        )                            where x=ω/ω0         and ω0=2π·1.023·106         
Note that the spectrum of the sine phase BOC(n,m) waveform, Hsinn,m(ω), consists entirely of imaginary components due to the presence of the j (=√−1) term in the denominator.
Similarly, the spectrum of a binary offset carrier, BOC(n,m), with a cosine phased spreading symbol modulation, is given in equation (3-1). Equation (3-1) shows the complex spectrum, Hcosn,m(ω), for values of (2n/m) that are even. This corresponds to (n/m) complete cycles of the binary offset carrier in each spreading code symbol. The complex spectrum is based on a calculation over the duration of a single code element, ΔT=2π/(mω0). The waveform used for the spectrum computations extends over the interval t∈(−ΔT/2, ΔT/2) and, for definition, has a positive dwell at t=0.
                                          H                          n              ,              m                        cos                    ⁡                      (            ω            )                          =                                            2              ⁢                                                          ⁢                              π                ·                                                      (                                          -                      1                                        )                                                        (                                                                  n                        m                                            +                      1                                        )                                                                                      m              ⁢                                                          ⁢                              ω                0                                              ·                                                    (                                  1                  -                                      cos                    ⁡                                          (                                                                        π                          ⁢                                                                                                          ⁢                          x                                                                          2                          ⁢                                                                                                          ⁢                          n                                                                    )                                                                      )                            ·                              sin                ⁡                                  (                                                            π                      ⁢                                                                                          ⁢                      x                                        m                                    )                                                                                    (                                                      π                    ⁢                                                                                  ⁢                    x                                    m                                )                            ·                              cos                ⁡                                  (                                                            π                      ⁢                                                                                          ⁢                      x                                                              2                      ⁢                                                                                          ⁢                      n                                                        )                                                                                        (                  3          ⁢                      -                    ⁢          1                )                            where x=ω/ω0         and ω0=2π·1.023·106         
Note that the spectrum of the cosine phased BOC(n,m) waveform, Hcosn,m(ω), consists entirely of real components.
The corresponding power spectral density (PSD) is given in equation (4) below and is averaged over 1 second assuming that each spreading code symbol takes a (binary) state selected randomly from the elements {+1,−1}. The PSD is:
                                          Φ                          n              ,              m                                ⁡                      (            x            )                          =                                            2              ⁢                                                          ⁢              π                                      m              ⁢                                                          ⁢                              ω                0                                              ·                                    {                                                                    sin                    ⁡                                          (                                                                        π                          ⁢                                                                                                          ⁢                          x                                                                          2                          ⁢                                                                                                          ⁢                          n                                                                    )                                                        ·                                      sin                    ⁡                                          (                                                                        π                          ⁢                                                                                                          ⁢                          x                                                m                                            )                                                                                                            {                                                                  π                        ⁢                                                                                                  ⁢                        x                                            m                                        }                                    ·                                      cos                    ⁡                                          (                                                                        π                          ⁢                                                                                                          ⁢                          x                                                                          2                          ⁢                                                                                                          ⁢                          n                                                                    )                                                                                  }                        2                                              (        4        )            
As discussed above, in a composite binary offset carrier (BOC) signal, as an alternative to time multiplexing, the signal is formed through the additive combining of two or more BOC components for each spreading symbol. Thus, each spreading symbol has a spectrum containing, for a 2 component case, a portion a of a BOC(n,m) component and a portion β of a BOC(k,m) component. Notice that both components have the same spreading code (chip) frequency (same duration of spreading code element). The composite complex spectrum, SC(ω), is then:SC(ω)=Hn,m(ω)+β·Hk,m(ω)   (5)
The corresponding power spectral density is formed from the product of SC(ω) with its complex conjugate, and for real α,β:
                                                                                          Φ                  c                                ⁡                                  (                  ω                  )                                            =                            ⁢                                                                    S                    C                    *                                    ⁡                                      (                    ω                    )                                                  ·                                                      S                    C                                    ⁡                                      (                    ω                    )                                                                                                                          =                            ⁢                                                                                                            S                      C                                        ⁡                                          (                      ω                      )                                                                                        2                                                                                        =                            ⁢                                                (                                                            α                      ·                                                                        H                                                      n                            ,                            m                                                    *                                                ⁡                                                  (                          ω                          )                                                                                      +                                          β                      ·                                                                        H                                                      k                            ,                            m                                                    *                                                ⁡                                                  (                          ω                          )                                                                                                      )                                ·                                  (                                                            α                      ·                                                                        H                                                      n                            ,                            m                                                                          ⁡                                                  (                          ω                          )                                                                                      +                                          β                      ·                                                                        H                                                      k                            ,                            m                                                                          ⁡                                                  (                          ω                          )                                                                                                      )                                                                                                        =                            ⁢                                                                    α                    2                                    ⁢                                                            Φ                                              n                        ,                        m                                                              ⁡                                          (                      ω                      )                                                                      +                                                      β                    2                                    ⁢                                                            Φ                                              k                        ,                        m                                                              ⁡                                          (                      ω                      )                                                                      +                                  α                  ⁢                                                                          ⁢                                      β                    ⁡                                          (                                                                                                    H                                                          n                              ,                              m                                                        *                                                    ·                                                      H                                                          k                              ,                              m                                                                                                      +                                                                              H                                                          n                              ,                              m                                                                                ·                                                      H                                                          k                              ,                              m                                                        *                                                                                              )                                                                                                                              (        6        )                                where        ⁢                                  ⁢                                            Φ                              n                ,                m                                      ⁡                          (              ω              )                                =                                                    H                                  n                  ,                  m                                *                            ⁡                              (                ω                )                                      ·                                          H                                  n                  ,                  m                                            ⁡                              (                ω                )                                                    ⁢                                  ⁢        and        ⁢                                  ⁢                                            Φ                              k                ,                m                                      ⁡                          (              ω              )                                =                                                    H                                  k                  ,                  m                                *                            ⁡                              (                ω                )                                      ·                                          H                                  k                  ,                  m                                            ⁡                              (                ω                )                                                                                    
Equation (6) clearly shows the differences in PSDs of the composite BOC (additive waveforms) and time multiplex approaches. The power spectral density, ΦTM(ω), for the time multiplex of BOC(n,m) and BOC(k,m) spreading symbol components, if the proportions are α2 and β2, is:ΦTM(ω)=α2·Φn,m(ω)+β2 Φk,m(ω)   (7)
Therefore, the time multiplex sequence comprises α2/(α2+β2) chips with a power spectral density of Φn,m(ω) and β2/(α2+β2) chips with a power spectral density of Φk,m(ω). The difference between the PSDs for CBOC and TMBOC techniques reside in the presence of the cross spectral terms in the CBOC PSD, Φcross(ω):Φcross(ω)=αβ(Hn,m*·Hk,m+Hn,m·Hk,m*)   (8)
The situation is exacerbated when, for example, there are 3 components forming the composite signal. In the time multiplex realisation, the components are interspersed amongst the code elements in suitable numbers to establish the contributory proportions required from each in the power spectral density to be transmitted (more correctly at the time of signal generation as there are transmission filters in the satellites that control out of band emissions). A typical example has the proportions α2,β2,δ2 for signals with each of three PSD's as equation (9) below illustrates.ΦTM(ω)=α2·Φn,m(ω)+β2·Φk,m(ω)+δ2·Φl,m(ω)   (9)
The corresponding spectrum for the additive method of producing a 3 component composite BOC signal has three cross spectral terms of the form of equation (9).
                                                                                          Φ                  C                                ⁡                                  (                  ω                  )                                            =                            ⁢                                                                    S                    C                    *                                    ⁡                                      (                    ω                    )                                                  ·                                                      S                    C                                    ⁡                                      (                    ω                    )                                                                                                                          =                            ⁢                                                (                                                            α                      ·                                                                        H                                                      n                            ,                            m                                                    *                                                ⁡                                                  (                          ω                          )                                                                                      +                                          β                      ·                                                                        H                                                      k                            ,                            m                                                    *                                                ⁡                                                  (                          ω                          )                                                                                      +                                          δ                      ·                                                                        H                                                      l                            ,                            m                                                    *                                                ⁡                                                  (                          ω                          )                                                                                                      )                                ·                                                                                                      ⁢                              (                                                      α                    ·                                                                  H                                                  n                          ,                          m                                                                    ⁡                                              (                        ω                        )                                                                              +                                      β                    ·                                                                  H                                                  k                          ,                          m                                                                    ⁡                                              (                        ω                        )                                                                              +                                      δ                    ·                                                                  H                                                  l                          ,                          m                                                                    ⁡                                              (                        ω                        )                                                                                            )                                                                                        =                            ⁢                                                                    α                    2                                    ⁢                                                            Φ                                              n                        ,                        m                                                              ⁡                                          (                      ω                      )                                                                      +                                                      β                    2                                    ⁢                                                            Φ                                              k                        ,                        m                                                              ⁡                                          (                      ω                      )                                                                      +                                                      δ                    2                                    ⁢                                                            Φ                                              l                        ,                        m                                                              ⁡                                          (                      ω                      )                                                                      +                                                                                                      ⁢                                                α                  ⁢                                                                          ⁢                                      β                    ⁡                                          (                                                                                                    H                                                          n                              ,                              m                                                        *                                                    ·                                                      H                                                          k                              ,                              m                                                                                                      +                                                                              H                                                          n                              ,                              m                                                                                ·                                                      H                                                          k                              ,                              m                                                        *                                                                                              )                                                                      +                                                                                                      ⁢                                                α                  ⁢                                                                          ⁢                                      δ                    ⁡                                          (                                                                                                    H                                                          n                              ,                              m                                                        *                                                    ·                                                      H                                                          l                              ,                              m                                                                                                      +                                                                              H                                                          n                              ,                              m                                                                                ·                                                      H                                                          l                              ,                              m                                                        *                                                                                              )                                                                      +                                                                                                      ⁢                              β                ⁢                                                                  ⁢                                  δ                  ⁡                                      (                                                                                            H                                                      k                            ,                            m                                                    *                                                ·                                                  H                                                      l                            ,                            m                                                                                              +                                                                        H                                                      k                            ,                            m                                                                          ·                                                  H                                                      l                            ,                            m                                                    *                                                                                      )                                                              ⁢                                                                                                      (        10        )                                where        ⁢                                  ⁢                                            Φ                              n                ,                m                                      ⁡                          (              ω              )                                =                                                    H                                  n                  ,                  m                                *                            ⁡                              (                ω                )                                      ·                                          H                                  n                  ,                  m                                            ⁡                              (                ω                )                                                    ⁢                                  ⁢                                            Φ                              k                ,                m                                      ⁡                          (              ω              )                                =                                                    H                                  k                  ,                  m                                *                            ⁡                              (                ω                )                                      ·                                          H                                  k                  ,                  m                                            ⁡                              (                ω                )                                                    ⁢                                  ⁢                                            Φ                              l                ,                m                                      ⁡                          (              ω              )                                =                                                    H                                  l                  ,                  m                                *                            ⁡                              (                ω                )                                      ·                                          H                                  l                  ,                  m                                            ⁡                              (                ω                )                                                                                    
The cross-spectral terms in equation (10) have a significant influence on the transmitted PSD. Clearly, the presence of the cross spectral terms hinders the realisation of a common PSD for CBOC and MBOC.
It is an object of embodiments of the present invention to at least mitigate one or more problems of the prior art.